01 25 17 09 33 06 35

20 16 28 04 12 15 31

26 07 29 18 10 34 02

21 23 13 32 05 24 08

22 19 03 27 30 11 14

11 31 16 06 15 26 21

17 27 07 22 32 12 09

13 03 28 18 08 23 33

29 24 04 14 34 19 02

20 05 35 30 01 10 25

13 09 05 35 33 07 24

23 10 30 02 31 08 22

15 16 17 18 19 20 21

14 27 06 34 04 29 12

25 28 32 01 03 26 11

05 27 15 12 25 19 23

32 03 16 29 14 28 04

06 35 26 18 10 01 30

34 08 22 07 20 33 02

13 17 11 24 21 09 31

The first two magic rectangles above were constructed on 7 November

and are rather irregular. The magic constants are 90 and 126 (i.e.

5x18 and 7x18).

The third is slightly more regular, having the sequence 15 to 21 along

the middle rank, similar to the method used for the 3x7 magic

rectangles reported here on 13 October.

The fourth is the most regular having diametrally opposite cells

complementary (i.e. adding to 36) except for two cases 32-04 and 34-

02. It may be that a completely symmetric magic tour is impossible,

but I've not been able to prove this so far. (It is possible on the

3x5 board as I showed in Chessics #26, 1986.)

Obviously every rank and file contains an even number of odd numbers,

and hence an odd number of even numbers. There are 8 numbers of the

form 4n, occurring in pairs of complements, and 9 of the form 4n + 2

consisting of the middle number 18 and four pairs of complements.

There remain 9 each of the forms 4n +1 and 4n + 3, which are

complements of each other.

The method I use for constructing these is to first put the numbers

into the array in some regular pattern, and then to adjust the rank

and file sums to give the magic values by making transpositions of

pairs or groups of numbers.

The ranks and files of these magic rectangles can be permuted without

affecting the magic property. So each generates 5!x7!/4 = 151200

(oriented with the long lines horizontal). The division by 4 is to

avoid counting rotations and reflections separately.

I've been trying to find an arrangement that uses the least number of

types of move but so far have not managed to reduce the number to less

than nine.

=====

Addendum 15 November:

A symmetric solution is possible. One is given in W.S.Andrews

*Magic Squares and Cubes*Fig.458 due to C.Planck.

Planck also seems to have anticipated my work on the 3x5 case,

since the 39 solutions are mentioned but not diagrammed.